Radar tracking is difficult, especially for air and missile targets—not only because of measurement errors, but also because of the wide variety of motion possible (with targets including, for example, helicopters, sub- and super-sonic aircraft, artillery, cruse and ballistic missiles). Track initiation constitutes a difficult part of tracking since only a minimal set of detections exist at the beginning of a track. Conventional methods for track-initiation tend not to work for this problem, as too few detections exist in one rotation period. Furthermore, such detections span too short a time interval to provide initial-tracks with a requisite predictive validity.
In general, air and missile defense radars use two types of antenna configurations: rotating antennas that cover the full 360-degrees of azimuth, but have a narrow instantaneous field of view, typically less than a few beamwidths wide; and electronically scanned antennas that do not rotate, but remain stationary and receive a much wider instantaneous field of view, which typically can cover a ninety-degree sector. A new class of antennas for air and missile defense radar is emerging that will rotate in azimuth while electronically scanning in both elevation and azimuth. This antenna technology has become cost-effective due to advancements in electronic components and the enhanced capabilities they offer. An antenna that is both rotating and is electronically scanned can provide the benefits of both the complete 360 degrees coverage and the wider electronically scanned instantaneous (but rotating) field of view. This advanced antenna technology, however, brings new challenges to radar systems, and in particular, in determining an initial-track of a target that has sufficient predictive validity using only one scan's detections.
To appreciate the problem realized and overcome by the present invention, its context within the data processing function of a radar system should be understood. In general, the primary functions of most air and missile defense radar systems are to detect and to report the presence of new threats (i.e. targets). A secondary function is to track those targets, which directly supports the surveillance function by reducing the probability of re-reporting a given target falsely as a new target the second and subsequent time it is detected. Between these two functions (the initial detection of a new target and tracking it) sits the sub-functions of verification and track initiation.
FIG. 1a-b illustrate operation of a prior art radar antenna that only rotates. Here the radar accomplishes surveillance and tracking using a common set of detections, which detections serve both to verify the existence of new targets and to determine their initial-tracks. FIG. 1a illustrates operation of an azimuthally rotating (no azimuthal electronic scan) wherein rotation of the antenna covers an area of 360 degrees in one scan, 100, with instantaneous coverage being one beamwidth, 102. Rotation of the antenna prevents the radar from determining initial-tracks using only one scan's worth of detections—during one rotation period of the antenna the target might be detected only once and hence the detection cannot be verified.
FIG. 1b illustrates the conventional method these radars use for track initiation, the so-called “track before detect” (TBD) method. A sequence of scans 1-4 (antenna rotations) first determines which detections in a given scan are not associated with any known track. Through processing of the detected data the system constructs a set of candidate initial-tracks from those unassociated detections and determines the best initial-track from among those candidate tracks. The TBD method accordingly accomplishes the two functions of target verification and track initiation concurrently, but requires several scans to complete. A relatively larger time between scans requires more than one detection be tested for possible verification.
FIG. 2a illustrates the case of a radar antenna that is fully electronically scanned but does not rotate. This antenna can provide data on a target virtually without interruption for as long as the target is within its much wider, albeit stationary field of view. Such non-rotating, electronically scanned (e.g. in azimuth and elevation) systems have larger instantaneous coverage 102′ but do not cover the full 360 degrees. As illustrated in FIG. 2b, such radar can proceed immediately from an initial unassociated detection to verification and then into track initiation. The conventional track initiation method used for a fully electronically scanned antenna that does not rotate is to immediately point a verification beam directly at an unassociated detection and attempt to re-detect the target (until a timeout) and, since the unknown target is possibly moving, place the next beam at the location of the previous detection to thereby collect a sequence of measurements on the moving target's position until an initial-track having sufficient predictive validity can be constructed. This radar accordingly accomplishes the two functions of target verification and track initiation sequentially.
As further background on the problem of track initiation, FIG. 3 provides an exemplary illustration of a radar detection in two dimensions, the XY-plane, of a target whose motion is defined by two independent and unknown functions of time x(t) and y(t). A radar located at the coordinate origin attempts to detect the target and to measure its position in terms of its range and azimuth, (r, a), relative to the radar. These components are similar to polar coordinates and accordingly are identified as the radar coordinates of the target. Basic trigonometry provides the relationships between Cartesian and radar coordinates: x=r sin a and y=r cos a. In FIG. 3 the true position of the target at the time of detection is denoted by the closed dot while the detection of the target by the radar is denoted in FIG. 3 by the symbol x. The detection provides noisy measurements of the radar coordinates, (R, A), which contain errors, (εR,εA), and so the relationships between the measured values of the radar coordinates and the unknown true ones are R=r+εR and A=a+εA. In the sequel all measurements will be presented as being unbiased and identically distributed with standard deviations σR and σA, which simplifies the notation, but the apparatuses, systems and methods presented here are valid for the general case where measurements are biased and have different variances.
Referring to FIG. 4, there is illustrated a two-dimensional confidence interval of the aforementioned noisy measurement of the target's range and azimuth. As is known by those skilled in the art, in two dimensions a confidence interval 400 is an ellipse, and in three dimensions it is an ellipsoid. A position confidence interval is where the true position of the target is found during a specified percentage of an interval of time. In FIG. 4 the true position of the target is the closed dot 410 while the measurement 420 is the “x” at the center of the ellipse.
Before determining the initial-track, given a new detection, the first problem the radar solves is that of confirming whether a detection proves to be a new target, as opposed to a detection that proves not to be a target referred to as a false alarm. Obtaining a second, independent detection close in time to the first detection is generally desired for this purpose. If the detections are sufficiently close in time, then the unknown motion of the target can be ignored for the purpose of verification. If, however, the time between the two detections is large, then the outcome may be inconclusive.
Referring to FIG. 5a there is illustrated two detections 510, 520 of the same target that are sufficiently close in time so that their position measurement confidence intervals overlap, which is then taken to imply the positive verification of the target's existence and not a false alarm. However, in FIG. 5b the two detections of the same target are not sufficiently close in time, in which case the unknown motion of the target manifests itself and defeats simplistic verification. Thus, there is ambiguity as to whether one target does indeed exist, or whether there are two (unverified) targets; or that there are two false alarms, or that one target exists and that there is a false alarm.
Computationally, simplistic target verification using two independent detections can be accomplished using a chi χ2 type test with a threshold γ2. For example, let (Ri, Ai) be the position measurement of the initial detection, (Rv, Av) be the position measurement of the verification detection, and let the variances of the respective measurement components be (σR2, σA2). Then the outcome of the conventional chi-squared verification method that the detections serve to verify the existence of the target, can be expressed algebraically as
                                                                        (                                                      R                    i                                    -                                      R                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                R                2                                              +                                                    (                                                      A                    i                                    -                                      A                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                A                2                                                    ≤                              γ            2                    .                                    Equation        ⁢                                  ⁢        1            
In contrast, outcome of the conventional chi-squared verification method that the detections do not serve to verify the existence of the target, is
                                                                        (                                                      R                    i                                    -                                      R                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                R                2                                              +                                                    (                                                      A                    i                                    -                                      A                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                A                2                                                    >                              γ            2                    .                                    Equation        ⁢                                  ⁢        2            
If two independent detections are deemed to be associated, and hence they verify the existence of a new target, they also can be used to determine its initial-track. For example, for either of the two sets of detections in FIG. 5a-b, the prior art method for determining the motion of the target is to average the position measurements and also to difference them. This, in turn, leads to an estimate of the position and velocity of the target (with more than two detections a method of least-squares is used). Thus if (Ri, Ai) is the position measurement of the initial detection at time ti and (Rv, Av) is the position measurement of the verification detection at time tv then the initial-track for the target's motion is
                              [                                                                      r                  ^                                                                                                                          r                    .                                    ^                                                                                                      a                  ^                                                                                                                          a                    .                                    ^                                                              ]                =                                            [                                                                                                                  (                                                                              R                            i                                                    +                                                      R                            v                                                                          )                                            /                      2                                                                                                                                                                                    (                                                                                    R                              i                                                        -                                                          R                              v                                                                                )                                                /                        Δ                                            ⁢                                                                                          ⁢                      T                                                                                                                                                          (                                                                              A                            i                                                    +                                                      A                            v                                                                          )                                            /                      2                                                                                                                                                                                    (                                                                                    A                              i                                                        -                                                          A                              v                                                                                )                                                /                        Δ                                            ⁢                                                                                          ⁢                      T                                                                                  ]                        ⁢                                                  ⁢            with            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢            T                    =                                                    t                i                            -                              t                v                                      ≠            0.                                              Equation        ⁢                                  ⁢        3            
In Equation 3 as well as in the sequel, an over-carrot, “^”, indicates that the symbol is an estimated quantity and not a true value. Also, in Equation 3 the estimated quantities have been arranged into the form of a “state vector”. This state vector is the initial-track of the target in radar coordinates. An alternative form for the state vector is to express it in Cartesian coordinates, ({circumflex over (x)},{dot over ({circumflex over (x)},ŷ,{dot over (ŷ)}). A state vector in radar coordinates is more statistically valid, but a state vector in Cartesian coordinates is more generally useful. The Cartesian state vector, however, is not unique. There are two paths available for determining its components from the radar measurements: either first transform the measurements from radar coordinates into Cartesian coordinates and then apply the sum and difference operations on them; or first determine the state vector in radar coordinates using Equation 3 and then transform that state vector into Cartesian (rectangular) coordinates—the prior art for these two paths to ({circumflex over (x)},{dot over ({circumflex over (x)},ŷ,{dot over (ŷ)}) are illustrated in FIG. 6, blocks 610, 620, 630, and 640. These two Cartesian state vectors are similar but are not equal since the transformation between radar and rectangular coordinates is non-linear.
Since the measurements have errors, the components of the state vector of an initial-track will also have errors. Specifically, in terms of the unknown true position of the target, (r, a), and the respective measurement errors of the position measurement, (σR2, σA2), the errors of the initial-track position are{circumflex over (r)}−r=(εR(ti)+εR(tv))/2â−a=(εA(ti)+εA(tv))/2.  Equation 4
In terms of the unknown true velocity of the target, ({dot over (r)}, {dot over (a)}), the errors of the initial-track position velocity of the initial-track are{dot over ({circumflex over (r)}−{dot over (r)}=(εR(ti)−εR(tv))/ΔT {dot over (â)}−{dot over (a)}=(εA(ti)−εA(tv))/ΔT with ΔT=ti−tv≠0.  Equation 5
For the variances of the components of the initial track given in Equation 4 and Equation 5 are (valid at the midpoint time, (ti+tv)/2, when the respective measurement errors are unbiased and identically distributed)var({circumflex over (r)})=σR2/2 and var({dot over ({circumflex over (r)})=2σR2/ΔT2 var(â)=σA2/2 and var({dot over (â)})=2σA2/ΔT2.  Equation 6
Now together Equation 1 and Equation 5 illustrate the essential challenge that this invention overcomes. Two (or more) detections of a target that are closely spaced in time for the purpose of verifying the existence of a new target will usually be too closely spaced in time for determining an initial-track that has sufficient predictive validity. This deficiency is caused by having to divide the differences of the position measurements by the small time interval they span, ΔT—the estimated velocity components become the difference of measurement errors divided by a small time increment. The estimated velocity (and acceleration if included) so determined has little or no predictive validity. Using a larger ΔT in Equation 3 mitigates the effects of measurement errors in the velocity (and acceleration if included) and thereby increases the predictive validity of the initial track. Accordingly, the conventional track-initiation methods seek to achieve predictive validity by waiting until ΔT is sufficiently large before determining the state vector: when using an antenna that only rotates one must wait for a detection in the next scan; when using an antenna that only stares one can keep placing verification/track-initiation beams at the previous detection until ΔT is sufficiently large.
For completeness and for subsequent reference, in contrast with Equation 1, the prior art for the verification test when the radar uses three degrees of freedom is
                                                                        (                                                      R                    i                                    -                                      R                    v                                                  )                            2                                      2              ⁢                              σ                R                2                                              +                                                    (                                                      E                    i                                    -                                      E                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                E                2                                              +                                                    (                                                      A                    i                                    -                                      A                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                A                2                                                    ≤                              γ            2                    ⁢                                          ⁢                                    (              verified              )                        .                                              Equation        ⁢                                  ⁢        7            
                                                                        (                                                      R                    i                                    -                                      R                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                R                2                                              +                                                    (                                                      E                    i                                    -                                      E                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                E                2                                              +                                                    (                                                      A                    i                                    -                                      A                    v                                                  )                            2                                      2              ⁢                                                          ⁢                              σ                A                2                                                    >                              γ            2                    ⁢                                          ⁢                                    (                              not                ⁢                                                                  ⁢                verified                            )                        .                                              Equation        ⁢                                  ⁢        8            
Furthermore, in contrast with Equation 3, the prior art operation when the radar uses a zero-order model (constant position (“CP”), assumption—the target is assumed to be stationary) with three degrees of freedom, the state vector of an initial track based on two detections, (Ri, Ei, Ai) and (Rv, Ev, Av) is three-dimensional, valid at the midpoint (t1+t2)/2:
                              [                                                                      r                  ^                                                                                                      e                  ^                                                                                                      a                  ^                                                              ]                =                              [                                                                                                      (                                                                        R                          i                                                +                                                  R                          v                                                                    )                                        /                    2                                                                                                                                          (                                                                        E                          i                                                +                                                  E                          v                                                                    )                                        /                    2                                                                                                                                          (                                                                        A                          i                                                +                                                  A                          v                                                                    )                                        /                    2                                                                        ]                    .                                    Equation        ⁢                                  ⁢        9            
When the radar uses a first order model (constant velocity (“CV”), assumption—the target is assumed to have linear motion) with three degrees of freedom, the state vector of an initial track based on two detections, (Ri, Ei, Ai) and (Rv, Ev, Av) is six-dimensional, valid at the midpoint (t1+t2)/2:
                              [                                                                      r                  ^                                                                                                                          r                    .                                    ^                                                                                                      e                  ^                                                                                                                          e                    .                                    ^                                                                                                      a                  ^                                                                                                                          a                    .                                    ^                                                              ]                =                                            [                                                                                                                  (                                                                              R                            i                                                    +                                                      R                            v                                                                          )                                            /                      2                                                                                                                                                                                    (                                                                                    R                              i                                                        -                                                          R                              v                                                                                )                                                /                        Δ                                            ⁢                                                                                          ⁢                      T                                                                                                                                                          (                                                                              E                            i                                                    +                                                      E                            v                                                                          )                                            /                      2                                                                                                                                                                                    (                                                                                    E                              i                                                        -                                                          E                              v                                                                                )                                                /                        Δ                                            ⁢                                                                                          ⁢                      T                                                                                                                                                          (                                                                              A                            i                                                    +                                                      A                            v                                                                          )                                            /                      2                                                                                                                                                                                    (                                                                                    A                              i                                                        -                                                          A                              v                                                                                )                                                /                        Δ                                            ⁢                                                                                          ⁢                      T                                                                                  ]                        ⁢                                                  ⁢            with            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢            T                    =                                                    t                i                            -                              t                v                                      ≠            0.                                              Equation        ⁢                                  ⁢        10            
When the radar uses a second order model (constant acceleration (“CA”), assumption) with three degrees of freedom, the state vector of an initial track based on three detections, (R1, E1, A1), (R2, E2, A2) and (R3, E3, A3 ), is nine-dimensional, valid at the midpoint t2=(t1+t3)/2 is:
                                                                                          [                                                                                                              r                          ^                                                                                                                                                                                          r                            .                                                    ^                                                                                                                                                                                          r                            ¨                                                    ^                                                                                                                                                              e                          ^                                                                                                                                                                                          e                            .                                                    ^                                                                                                                                                                                          e                            ¨                                                    ^                                                                                                                                                              a                          ^                                                                                                                                                              a                          .                                                                                                                                                                                          a                            ¨                                                    ^                                                                                                      ]                                =                                                      [                                                                                                                        R                            2                                                                                                                                                                                                                                          (                                                                                                      R                                    1                                                                    -                                                                      R                                    3                                                                                                  )                                                            /                              2                                                        ⁢                                                                                                                  ⁢                            Δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                                                                                                                                          (                                                                                                      R                                    1                                                                    -                                                                      2                                    ⁢                                                                          R                                      2                                                                                                        +                                                                      R                                    3                                                                                                  )                                                            /                              Δ                                                        ⁢                                                                                                                  ⁢                                                          T                              2                                                                                                                                                                                                        E                            2                                                                                                                                                                                                                                          (                                                                                                      E                                    1                                                                    -                                                                      E                                    3                                                                                                  )                                                            /                              2                                                        ⁢                                                                                                                  ⁢                            Δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                                                                                                                                          (                                                                                                      E                                    1                                                                    -                                                                      2                                    ⁢                                                                          E                                      2                                                                                                        +                                                                      E                                    3                                                                                                  )                                                            /                              Δ                                                        ⁢                                                                                                                  ⁢                                                          T                              2                                                                                                                                                                                                        A                            2                                                                                                                                                                                                                                          (                                                                                                      A                                    1                                                                    -                                                                      A                                    3                                                                                                  )                                                            /                              2                                                        ⁢                                                                                                                  ⁢                            Δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                                                                                                                                          (                                                                                                      A                                    1                                                                    -                                                                      2                                    ⁢                                                                          A                                      2                                                                                                        +                                                                      A                                    3                                                                                                  )                                                            /                              Δ                                                        ⁢                                                                                                                  ⁢                                                          T                              2                                                                                                                                            ]                                    ⁢                                                                          ⁢                  with                                            ⁢                                                                                                                                      Δ                ⁢                                                                  ⁢                T                            =                                                                    t                    i                                    -                                      t                    2                                                  =                                                                            t                      2                                        -                                          t                      3                                                        ≠                  0.                                                                                        Equation        ⁢                                  ⁢        11            
For the class of radars having a rotating electronically scanned antennas, the simple problem of verifying the existence of a new target is essentially the same as that faced by a conventional staring, electronically scanned radar antenna, which is to associate independent detections that are closely spaced in time. Use of the conventional algorithms requires either several scans of a rotating antenna or more staring time of a non-rotating electronically scanned antenna, neither of which are available during a single scan of this antenna, which concurrently rotates and is electronically scanned.
An alternative method and system that overcomes the above-identified problems, including for example, determining that the initial-track that has sufficient predictive validity using only the detections of a target that are obtained in one rotation of its electronically scanned antenna, is highly desired.